Simplify- xaxba2-ab -b2-xbxcb2-bc-c2-xcxac2-ca-a2

(x^ax^b)^a^2+ab +b^2×(x^bx^c)^b^2+bc+c^2×(x^cx^a)^c^2+ca+a^2 =(x^a b)^a^2+ab+b^2× (x^b c)^b^2+bc+c^2× (x^c a)^c^2+ca+a^2 =x^a^3 b^3× x^b^3 c^3× x ...

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Byju's Answer

Standard IX

Mathematics

Exponential Representation for Irrational Numbers

Simplify: xa...

Question

Simplify:
${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}$

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Solution

${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}$
$= (x^{a - b})^{a^{2} + a b + b^{2}} \times (x^{b - c})^{b^{2} + b c + c^{2}} \times (x^{c - a})^{c^{2} + c a + a^{2}}$
$= x^{a^{3} - b^{3}} \times x^{b^{3} - c^{3}} \times x^{c^{3} - a^{3}}$
$x^{a^{3} - b^{3} + b^{3} - c^{3} + c^{3} - a^{3}}$
$x^{0}$
$= 1$

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Similar questions

Prove that :

(i) ${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + a b + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1$

(ii) ${(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1$

Q. Prove that:

(i)

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}} = 1

(ii)

{(\frac{x^{a}}{x^{- b}})}^{a^{2} - a b + b^{2}} \times {(\frac{x^{b}}{x^{- c}})}^{b^{2} - b c + c^{2}} \times {(\frac{x^{c}}{x^{- a}})}^{c^{2} - c a + a^{2}} = 1

(iii)

{(\frac{x^{a}}{x^{b}})}^{c} \times {(\frac{x^{b}}{x^{c}})}^{a} \times {(\frac{x^{c}}{x^{a}})}^{b} = 1

Q. Simplify :

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}

Q. The value of

{(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \cdot {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \cdot {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + b^{2}}

State true or false

Simplified form of

(\frac{x^{a}}{x^{b}})^{a^{2} + a b + b^{2}} \times (\frac{x^{b}}{x^{c}})^{b^{2} + b c + c^{2}} \times (\frac{x^{c}}{x^{a}})^{c^{2} + c a + a^{2}}

is 1

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Simplify:(xaxb)a2+ab+b2×(xbxc)b2+bc+c2×(xcxa)c2+ca+a2

(xaxb)a2+ab+b2×(xbxc)b2+bc+c2×(xcxa)c2+ca+a2=(xa−b)a2+ab+b2×(xb−c)b2+bc+c2×(xc−a)c2+ca+a2=xa3−b3×xb3−c3×xc3−a3xa3−b3+b3−c3+c3−a3x0=1

Simplify:
${(\frac{x^{a}}{x^{b}})}^{a^{2} + a b + b^{2}} \times {(\frac{x^{b}}{x^{c}})}^{b^{2} + b c + c^{2}} \times {(\frac{x^{c}}{x^{a}})}^{c^{2} + c a + a^{2}}$